1. Field of the Invention
In the realm of 3-D migration of seismic data, this invention is concerned with the correction of a velocity independent dip-moveout (DMO) operator for the effects of earth inhomogeneities such as transverse anisotropy and ray bending due to a variation in wavefield propagation velocity as a function of depth.
2. Related Art
As is well known, DMO is a process for collapsing non-zero offset seismic wavefield ray paths to zero-offset normally-incident ray paths. DMO compensates for dip- and azimuth-dependent stacking velocities. In the presence of a constant velocity medium, the DMO operator is independent of velocity, it is two dimensional and it is oriented along a line between the source and receiver. In the presence of vertical, lateral and angular velocity changes, singly or in combination, the DMO operator becomes a three-dimensional complex surface that is velocity-dependent.
Various authors have addressed the problem by constructing a 3-D DMO operator from isochronal surfaces. An isochronal surface defines the shape of a reflector that will give the same arrival time at a fixed location in a common offset section. The required surface is found by ray-tracing. Having found the requisite surface, the DMO operator is created by performing a modeling experiment assuming coincident source-receiver locations. This and other similar techniques tend to compensate the DMO operator for refraction effects produced by velocity variations. They do not address the problem of conflict of migrated and unmigrated times. That problem arises because velocities for DMO processing need to be defined at the migrated time and location of the reflectors but because during DMO processing the data have not been migrated the energy of the reflectors is located at their unmigrated positions. That creates a velocity conflict among reflectors from different dips and azimuths. A further problem is anisotropy wherein the stacking velocity is different for different azimuths and dips.
Deregowski and Rocca showed that a velocity-independent DMO operator for a common offset section can be expressed as EQU t.sub.1 =t.sub.0 (1-(x.sup.2 /h.sup.2)).sup.1/2, (1)
where t.sub.1 is the zero-offset time, t.sub.0 is the normal moveout (NMO) corrected time, x is the distance along the midpoint section and h is half the source-to-receiver offset. Equation (1) defines a velocity-independent 2-D DMO operator that moves energy in a vertical plane along the source-to-receiver direction. Formulation (1) may be extended to 3-D but only provided a constant-velocity earth exists. It is important to understand that the DMO operator does not attempt to move energy to its final image position. That step is performed by the migration process after stacking. DMO is merely a partial migration to zero offset before stack.
For an inhomogeneous earth wherein the velocity is a function of depth, a 2-D operator may be found which is necessarily velocity dependent. A velocity-dependent 2-D DMO operator is given by EQU t.sub.1 =t.sub.0 (1-(x.sup.2 /.gamma..sup.2 h.sup.2)).sup.1/2,(2)
where EQU .gamma.=(1-(t.sub.0 /v)(dv/dt.sub.0)).sup.1/2, (3)
In a three dimensional inhomogeneous earth, because of ray bending, a 3-D DMO operator becomes a very complicated surface such that expensive ray tracing is necessary.
Rothman et al. introduced the concept of residual migration or the inverse operation which is sometimes referred to as modeling. Their process is valid for a constant velocity but may be extended to the case of a variable velocity. Residual migration is the process of recovering the correct image of the earth after the data have been post-stack migrated with an incorrect velocity. The correct image of the subsurface can be recovered by re-migrating the output of the first migration but using a residual velocity. Migration is a process that is applied when the first velocity is too low; modeling (inverse migration) is a process that is applied when the first velocity was too high. Thus residual migration is a process that is applied to the data but only after DMO has been applied and after the data have been stacked. The step of creating and applying a DMO operator in the presence of an inhomogeneous earth is not the same as the step of performing residual migration.
Rothman et al. did not explore extensions of their concept of residual migration to a 3-D DMO operator. However, Hale and Artley teach construction of a 2-D DMO operator, for use in the presence of a vertical velocity gradient, in their paper "Squeezing dip moveout for depth-variable velocity" in Geophysics, v. 58, n. 2, February 1993, p. 257. Their DMO algorithm is valid for 2-D but not for 3-D; anisotropy remains unaccounted for.
For purposes of this disclosure, the unqualified term "velocity" means the propagation velocity of an acoustic wavefield through elastic media. There is a need for a method for efficiently constructing a velocity-dependent 3D DMO operator that does not require classical ray tracing methods.
This invention fills that need by first processing and sorting the data into common offset gathers and, if needed, additionally sorting into common azimuth gathers. Normal moveout is applied to the sorted data based on the average RMS velocity to flat-lying beds. An intermediate data set is formed by applying a velocity-independent DMO operator to the normal-moveout-corrected common offset gathers according to (1). A residual 3-D DMO operator is defined and applied to the intermediate data set to compensate for inhomogeneities in the subsurface of the earth thereby to create a zero-offset data set. The so-compensated zero-offset data sets are resorted to common midpoint gathers for final processing including stacking and migration.